Thinking About Math

It's Wednesday night, and A10 Jadwin is filling up
fast.Well before the speaker is scheduled to start, there's
not a seat to be had; nearly 200 students, professors,
professionals and even local highschool students cram
themselves into every bit of floor space.

A large man with a wiry gray beard and a T-shirt printed
with a design by M.C. Escher walks to the front of the hall:
John Conway, John Von Neumann Professor in Applied and
Computational Mathematics. With a brief quip about how he
wasn't sure he'd have an audience, he begins to discuss the
ancient Greek mathematician Archimedes.

The sole surviving record of Archimedes' work in the
original Greek is a palimpsest, a papyrus document that was
written on, erased, then turned and used again. Modern
technology enables scholars to puzzle out the bottom layer
of text underneath the later writing.

Conway has made a career of finding new layers of meaning
in physical things. His fascination with games and puzzles
has led him to important discoveries in areas of mathematics
from number theory to geometry. He has broken new ground
with theories about knots and sphere packing, and discovered
one of the largest finite groups of numbers (known as the
Conway group).

"Thinking about Math (and Many Other Things)"On mathematicians
in history: "As mathematicians we are unique
among the sciences (with the slight exception of
physics) because our history goes back so far.
Those Greek geometers were doing amazing stuff
2,500 years ago." Conway hired an artist he met in
a café to paint a frieze around his office
depicting mathematicians through history. In
talking with students, he likes to gesture to the
paintings. "It adds something to see the man
looking down at you."On discovering the surreal
numbers: "For about six weeks, I went around in
a permanent sense of daydream." During a recent
trip to Australia, he began to think the sensation
was something like "what the famous explorers felt
when they stood on the edge of a vast new
landscape. In a way, discovering the surreal
numbers is like discovering a whole new continent.
There's a world that no one has seen before. Of
course, it's not the same. The surreal numbers are
not a physical thing. On the other hand you can
carry the concept around in your head, which you
can't do with Australia."On the future of the surreal
numbers: "One thing that worries me is that
there aren't any questions about the real numbers
that remain to be solved. The best math usually
turns out to have applications in mathematics or
physics. My numbers haven't found any application
at all yet, and that disappoints me. But in respect
to the surreal numbers, I am very proud. They may
just be a footnote to mathematics, but they are a
real, permanent footnote. And in 2000 years, it
will be possible for people to look them up and
think about them. There is eternality about math
questions, a feeling that you're really getting to
bottom of something."

His interest in the game Go helped him to identify an
entirely new class of numbers, called the surreal numbers.
In the nonacademic world he is known as the inventor of the
game Life, in which checker-like objects replicate
themselves and form patterns on a grid. The game, in turn,
has spawned a field of research called cellular
automata.

Conway is passionate about connecting people to math,
which is one reason he's doing the public lecture series in
Jadwin Hall.

"I really love my subject, and I want to spread it as
widely as I can, to spread the gospel," he says.

The series of eight lectures is titled "Thinking about
Math (and Many Other Things)."

The series has drawn students and faculty not only from
the math, physics and engineering departments, but also from
such areas as philosophy, linguistics, history and Hellenic
studies.

Conway is "a great lecturer," said Dimitri Gondicas,
executive director of the Program in Hellenic Studies. "He
brings math alive and makes it relevant, and also puts it in
its historical and philosophical context."

"He seems to know just how far to go into technical
details and still keep his audience interested, and also
convey the mathematical truths. That's a very delicate
balance," said philosophy professor John Burgess, who
attended with his son Fokion, a freshman at Princeton High
School.

Proofs, historical tidbits, asides

In his lecture on Archimedes, Conway wove together the
nuances of the ancient Greek's mathematical proofs with
historical tidbits and asides about his own favorite books.
He explained that he has been enamored of Archimedes ever
since he read his proof of the formula for the perimeter of
a circle.

"When I saw these theorems," he said, "my feeling was
'How is he going to fudge it?' I knew it was impossible,
absolutely impossible, to prove this from Euclid's
postulates. So I turned a few pages and got a spinetingling
feelingbecause Archimedes didn't fudge it, not one little
bit."

The palimpsest, which contains six books by Archimedes,
was originally written in the 12th century. (Religious texts
were written over the math a century later.) It is the only
text of Archimedes in Greek, which is especially important,
Conway noted, because the extant Latin translations were
done by scholars who lacked a profound understanding of
math.

When Conway found out that the palimpsest was going to be
sold at Christie's Auction House last fall, he lobbied for
Princeton to buy it. He convinced the University to put in a
bid for $1.2 million, but the document sold for more than $2
million to a private collector. Fortunately, Conway said,
the new owner has pledged to make the document available for
study.

Despite the emphasis on history in his public lectures,
Conway's real goal is to convey the most fascinating aspects
of the math itself. His second lecture addressed the work of
Kurt Gödel and the concept of uncertainty; the third
focused on the set theory and infinite numbers of Georg
Cantor; and the fourth turned to Conway's own surreal
numbers. The topic of the next lecture, at 8:00 pm on
November 17, is "Geometry, Logic and Physics."

"Syracuse was the math capital of the Greek world;
Princeton is the math capital of the modern world," quipped
physics professor Kirk McDonald at one lecture. "I grew up
in Arizona, and you'd hear about these great mathematicians
on the East Coast, and occasionally you'd hear about this
guy John Conway. And now here we are actually hearing
him."